We show that the noncommutative Wess-Zumino model is renormalizable to all orders of perturbation theory. The noncommutative scalar potential by itself is non-renormalizable but the Yukawa terms demanded by supersymmetry improve the situation turning the theory into a renormalizable one. As in the commutative case, there are neither quadratic nor linear divergences.Hence, the IR/UV mixing does not give rise to quadratic infrared poles.

For systems possessing only first-class constraints, we rigorously prove that the secondary constraints act as independent generators of gauge transformations (Dirac's conjecture). The proof essentially consists in demonstrating that the total and the extended Hamiltonian generate the same time evolution for the canonical realization, f(q,p), of the gauge-invariant quantities. We also dis-.cuss the alternative realization of these quantities.

We study the quantum dynamics of chiral fermion fields minimally coupled to a gauge field. The model, originally proposed by Jackiw and Rajaraman, is known to exhibit the anomalous breaking of gauge invariance, which leads to the appearance of an arbitrary parameter a. Both functional and operator techniques are used to obtain the two-point fermion Green's functions for a~1 and a =1. In both cases clustering holds, and the theory contains asymptotically free fermions. The quantum equation of motion for the field tensor resembles formally that of the Proca theory, but with a dynamically generated mass and a nonconserved source. It is found that for a =1 the generating functional cannot be written in terms of a manifestly Lorentz-invariant Lagrangian.

We study the modifications induced by spacetime anisotropy on the Casimir effect in the case of two parallel plates. Nonperturbative and perturbative regimes are analyzed. In the first case the Casimir force either vanishes or it reverses its direction which, in any case, makes the proposal untenable. On the other hand, the perturbative model enables us to incorporate appropriately the effects of spacetime anisotropy.

We show that the noncommutativity of space-time destroys the renormalizability of the 1/N expansion of the O(N ) Gross-Neveu model. A similar statement holds for the noncommutative nonlinear sigma model. However, we show that, up to the subleading order in 1/N expansion, the noncommutative supersymmetric O(N ) nonlinear sigma model becomes renormalizable in D = 3. We also show that dynamical mass generation is restored and there is no catastrophic UV/IR mixing. Unlike the commutative case, we find that the Lagrange multiplier fields, which enforce the supersymmetric constraints, are also renormalized. For D = 2 the divergence of the four point function of the basic scalar field, which in D = 3 is absent, cannot be eliminated by means of a counterterm having the structure of a Moyal product.